Average Error: 20.0 → 0.3
Time: 17.2s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x}}}{x + 1} \cdot \frac{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x}}}{x + 1} \cdot \frac{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}
double f(double x) {
        double r5394209 = 1.0;
        double r5394210 = x;
        double r5394211 = sqrt(r5394210);
        double r5394212 = r5394209 / r5394211;
        double r5394213 = r5394210 + r5394209;
        double r5394214 = sqrt(r5394213);
        double r5394215 = r5394209 / r5394214;
        double r5394216 = r5394212 - r5394215;
        return r5394216;
}
↓
double f(double x) {
        double r5394217 = 1.0;
        double r5394218 = x;
        double r5394219 = sqrt(r5394218);
        double r5394220 = r5394217 / r5394219;
        double r5394221 = r5394218 + r5394217;
        double r5394222 = r5394220 / r5394221;
        double r5394223 = sqrt(r5394221);
        double r5394224 = r5394217 / r5394223;
        double r5394225 = r5394220 + r5394224;
        double r5394226 = r5394220 / r5394225;
        double r5394227 = r5394222 * r5394226;
        return r5394227;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
  2. Using strategy rm

  3. Applied flip--20.1

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
  4. Using strategy rm

  5. Applied frac-times25.2

    \[\leadsto \frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

  6. Applied frac-times20.1

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} - \frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

  7. Applied frac-sub19.9

    \[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) - \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(1 \cdot 1\right)}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

  8. Simplified19.6

    \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right) - x}}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

  9. Simplified19.6

    \[\leadsto \frac{\frac{\left(1 + x\right) - x}{\color{blue}{\left(1 + x\right) \cdot x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity19.6

    \[\leadsto \frac{\frac{\left(1 + x\right) - x}{\left(1 + x\right) \cdot x}}{\color{blue}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}}\]

  12. Applied *-un-lft-identity19.6

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left(1 + x\right) - x\right)}}{\left(1 + x\right) \cdot x}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]

  13. Applied times-frac19.6

    \[\leadsto \frac{\color{blue}{\frac{1}{1 + x} \cdot \frac{\left(1 + x\right) - x}{x}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)}\]

  14. Applied times-frac19.6

    \[\leadsto \color{blue}{\frac{\frac{1}{1 + x}}{1} \cdot \frac{\frac{\left(1 + x\right) - x}{x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]

  15. Simplified19.6

    \[\leadsto \color{blue}{\frac{1}{1 + x}} \cdot \frac{\frac{\left(1 + x\right) - x}{x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

  16. Simplified0.4

    \[\leadsto \frac{1}{1 + x} \cdot \color{blue}{\frac{\frac{1 + 0}{x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}}\]
  17. Using strategy rm

  18. Applied *-un-lft-identity0.4

    \[\leadsto \frac{1}{1 + x} \cdot \frac{\frac{1 + 0}{x}}{\color{blue}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)}}\]

  19. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{1}{1 + x} \cdot \frac{\frac{1 + 0}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)}\]

  20. Applied *-un-lft-identity0.5

    \[\leadsto \frac{1}{1 + x} \cdot \frac{\frac{\color{blue}{1 \cdot \left(1 + 0\right)}}{\sqrt{x} \cdot \sqrt{x}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)}\]

  21. Applied times-frac0.3

    \[\leadsto \frac{1}{1 + x} \cdot \frac{\color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{1 + 0}{\sqrt{x}}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)}\]

  22. Applied times-frac0.3

    \[\leadsto \frac{1}{1 + x} \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt{x}}}{1} \cdot \frac{\frac{1 + 0}{\sqrt{x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}\right)}\]

  23. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(\frac{1}{1 + x} \cdot \frac{\frac{1}{\sqrt{x}}}{1}\right) \cdot \frac{\frac{1 + 0}{\sqrt{x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}}\]

  24. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}}}{x + 1}} \cdot \frac{\frac{1 + 0}{\sqrt{x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}\]

  25. Final simplification0.3

    \[\leadsto \frac{\frac{1}{\sqrt{x}}}{x + 1} \cdot \frac{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))